On the (Non)Hadamard Property of the SJ State in a $1+1$D Causal Diamond

TL;DR

The softened SJ state is studied, which is a slight modification of the original state to make it Hadamard, to investigate whether some peculiar features of entanglement entropy in causal set theory may be linked to its non-Hadamard nature.

Abstract

The Sorkin-Johnston (SJ) state is a candidate physical vacuum state for a scalar field in a generic curved spacetime. It has the attractive feature that it is covariantly and uniquely defined in any globally hyperbolic spacetime, often reflecting the underlying symmetries if there are any. A potential drawback of the SJ state is that it does not always satisfy the Hadamard condition. In this work, we study the extent to which the SJ state in a $1+1$D causal diamond is Hadamard, finding that it is not Hadamard at the boundary. We then study the softened SJ state, which is a slight modification of the original state to make it Hadamard. We use the softened SJ state to investigate whether some peculiar features of entanglement entropy in causal set theory may be linked to its non-Hadamard nature.

Figures (16)

• Figure 1: Pairs of points $x$ and $x'$ that satisfy the divergence conditions in \ref{['eqn:divergence_conditions']}. These conditions are satisfied when both points are on the two neighbouring left or right edges of the boundary of the diamond. The two pairs of red points satisfy the condition $u-v'=\pm 2L$, and the two pairs of blue points satisfy the condition $v-u'=\pm2L$. In the coincidence limit when both points lie on either the left or right corner, both conditions are satisfied.
• Figure 2: The approximate errors at the $1^{st}$ and $2^{nd}$ orders of the series, $\epsilon^{(1)}$ and $\epsilon^{(2)}$ in the coincidence limit $u'\rightarrow u$, $v'\rightarrow v$. The plots demonstrate that these functions are finite in the full domain of the causal diamond.
• Figure 3: The top two plots show the maximum absolute value of the real part of $\epsilon^{(1)}$ and $\epsilon^{(2)}$, when evaluated on pairs of points at different "radii" $r_{(u,v)}$ and $r_{(u',v')}$, corresponding to the $(u,v)$ and $(u',v')$ coordinates shown in the bottom plot.
• Figure 4: $\epsilon^{(1)}_{,u}$ and $\epsilon^{(2)}_{,u}$ when $v^\prime = 0.5$. Divergences occur only for Re[$\epsilon^{(1)}_{,u}$] when $u = \pm L$ and $u'\neq\mp L$. $L$ is set to 1.
• Figure 5: The first three orders of $\epsilon_{,u}$ in the coincidence limit. A divergence is observed along $u = \pm L$ in the real part of $\epsilon^{(1)}_{,u}$.